We first show that \(\Gamma _{\textrm {OT}}\)  is strongly connected. Let \((M_1,\xi _1)\)  and \((M_2,\xi _2)\)  be two overtwisted contact manifolds. By [1]}, there exist a directed path \(p\)  from \((M_1,\xi _1)\)  to \((M_2,\xi _2)\)  in \(\Gamma \) . We argue that any vertex in \(p\)  corresponds to an overtwisted contact manifold. Let us assume the contrary. Since \((M_2,\xi _2)\)  is overtwisted, there exists an overtwisted contact manifold \((M_{\textrm {OT}},\xi _{\textrm {OT}})\)  that can be obtained by contact \((-1)\) -surgery from a tight contact manifold \((M_{\textrm {tight}},\xi _{\textrm {tight}})\)  contradicting Wand's result which says that contact \((-1)\) -surgery preserves tightness [2]}.
Next, we consider \(\Gamma _*\)  with \(*=\textrm {tight}\) , \(\textrm {Stein}\) , \(\textrm {strong}\) , \(\textrm {weak}\)  or \(\textrm {c}\ne 0\) . To show that \(\Gamma _*\)  is connected we show that there exist an undirected path in \(\Gamma _*\)  from \((S^3,\xi _{\mathrm {st}})\)  to any other contact manifold \((M,\xi )\)  with property \(*\) . We consider the contact \((+1)\) -surgery along the Legendrian unknot with Thurston–Bennequin invariant \(\operatorname{tb}=-2\)  and rotation number \(\operatorname{rot}=1\)  in \((S^3,\xi _{\mathrm {st}})\) . It is well-known that the resulting contact manifold is the overtwisted contact structure \(\xi _1\)  on \(S^3\)  with normalized \(d_3\) -invariant equal to 1 [3]}. Then by [1]}, there exists a directed path in \(\Gamma \)  of contact \((-1)\) -surgeries from \((S^3,\xi _1)\)  to \((M,\xi )\) . However, this path runs through at least one overtwisted contact manifold and hence it is not in \(\Gamma _*\) . In total we get a surgery link \(L\)  in \((S^3,\xi _{\mathrm {st}})\)  describing \((M,\xi )\)  with only a single contact surgery coefficient \((+1)\) . Now we change the order of the surgeries and first perform all contact \((-1)\) -surgeries and at the end we perform the single contact \((+1)\) -surgery. Since contact \((-1)\) -surgery is known to preserve any of the properties \(*\)  by [2]}, [6]}, [7]}, [8]}, we get a path in \(\Gamma _*\)  from \((S^3,\xi _{\mathrm {st}})\)  to \((M,\xi )\) .
The contact surgery subgraphs \(\Gamma _{\textrm {tight}}\) , \(\Gamma _{\textrm {strong}}\) , \(\Gamma _{\textrm {weak}}\)  and \(\Gamma _{\textrm {c}\ne 0}\)  are not strongly connected since there exists in each of this graphs a contact manifold \((M,\xi )\)  which is not Stein fillable [9]}, [10]} and then there cannot be a directed path from \((S^3,\xi _{\mathrm {st}})\)  to \((M,\xi )\) .
Finally we show that \(\Gamma _{\textrm {Stein}}\)  is not strongly connected. We show that there exists no directed path of contact \((-1)\) -surgeries from \((S^3,\xi _{\mathrm {st}})\)  to \((S^1\times S^2,\xi _{\mathrm {st}})\) . Let us assume the contrary. Then we get a simply-connected Stein cobordism from \((S^3,\xi _{\mathrm {st}})\)  to \((S^1\times S^2,\xi _{\mathrm {st}})\) . We glue this Stein cobordism to the standard 4-ball filling of \((S^3,\xi _{\mathrm {st}})\)  to get a simply connected Stein filling \((W,\omega _{\mathrm {st}})\)  of \((S^1\times S^2,\xi _{\mathrm {st}})\) . However, by [6]} any Stein filling of \((S^1\times S^2,\xi _{\mathrm {st}})\)  is diffeomorphic to \(S^1\times D^3\)  which is not simply-connected.
